Square Root of 228
2026-02-28 08:22 Diff
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Last updated on August 5, 2025

If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The square root is used in the field of vehicle design, finance, etc. Here, we will discuss the square root of 228.

What is the Square Root of 228?

The square root is the inverse of the square of the number. 228 is not a perfect square. The square root of 228 is expressed in both radical and exponential form. In the radical form, it is expressed as √228, whereas (228)^(1/2) in the exponential form. √228 ≈ 15.0997, which is an irrational number because it cannot be expressed in the form of p/q, where p and q are integers and q ≠ 0.

Finding the Square Root of 228

The prime factorization method is used for perfect square numbers. However, the prime factorization method is not used for non-perfect square numbers where the long-division method and approximation method are used. Let us now learn the following methods:

  • Prime factorization method
  • Long division method
  • Approximation method

Square Root of 228 by Prime Factorization Method

The product of prime factors is the prime factorization of a number. Now let us look at how 228 is broken down into its prime factors.

Step 1: Finding the prime factors of 228 Breaking it down, we get 2 x 2 x 3 x 19: 2^2 x 3^1 x 19^1

Step 2: Now we found out the prime factors of 228. The second step is to make pairs of those prime factors. Since 228 is not a perfect square, therefore the digits of the number can’t be grouped in pairs. Therefore, calculating 228 using prime factorization is not straightforward.

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Square Root of 228 by Long Division Method

The long division method is particularly used for non-perfect square numbers. In this method, we should check the closest perfect square number for the given number. Let us now learn how to find the square root using the long division method, step by step.

Step 1: To begin with, we need to group the numbers from right to left. In the case of 228, we need to group it as 28 and 2.

Step 2: Now we need to find n whose square is 2. We can say n as '1' because 1 x 1 is lesser than or equal to 2. Now the quotient is 1; after subtracting 1 from 2, the remainder is 1.

Step 3: Now let us bring down 28, which is the new dividend. Add the old divisor with the same number, 1 + 1, and we get 2, which will be our new divisor.

Step 4: The new divisor will be the sum of the dividend and quotient. Now we get 2n as the new divisor; we need to find the value of n.

Step 5: The next step is finding 2n x n ≤ 128. Let us consider n as 5; now 25 x 5 = 125.

Step 6: Subtract 128 from 125; the difference is 3, and the quotient is 15.

Step 7: Since the dividend is less than the divisor, we need to add a decimal point. Adding the decimal point allows us to add two zeroes to the dividend. Now the new dividend is 300.

Step 8: Now we need to find the new divisor that is 150 because 1500 x 2 = 3000.

Step 9: Subtracting 3000 from 3000, we get the result 0.

Step 10: Now the quotient is 15.0.

Step 11: Continue doing these steps until we get two numbers after the decimal point. Suppose if there are no decimal values, continue till the remainder is zero. So the square root of √228 ≈ 15.10

Square Root of 228 by Approximation Method

The approximation method is another method for finding square roots. It is an easy method to find the square root of a given number. Now let us learn how to find the square root of 228 using the approximation method.

Step 1: Now we have to find the closest perfect square of √228. The smallest perfect square less than 228 is 225, and the largest perfect square more than 228 is 256. √228 falls somewhere between 15 and 16.

Step 2: Now we need to apply the formula that is (Given number - smallest perfect square) / (Greater perfect square - smallest perfect square). Going by the formula (228 - 225) ÷ (256 - 225) = 3/31 ≈ 0.0968 Using the formula, we identified the decimal point of our square root. The next step is adding the value we got initially to the decimal number, which is 15 + 0.0968 ≈ 15.10, so the square root of 228 is approximately 15.10.

Common Mistakes and How to Avoid Them in the Square Root of 228

Students do make mistakes while finding square roots, like forgetting about the negative square root or skipping long division methods. Now let us look at a few of those mistakes that students tend to make in detail.

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Problem 1

Can you help Max find the area of a square box if its side length is given as √200?

Okay, lets begin

The area of the square is 200 square units.

Explanation

The area of the square = side².

The side length is given as √200.

Area of the square = side² = √200 x √200 = 200.

Therefore, the area of the square box is 200 square units.

Well explained 👍

Problem 2

A square-shaped building measuring 228 square feet is built; if each of the sides is √228, what will be the square feet of half of the building?

Okay, lets begin

114 square feet

Explanation

We can just divide the given area by 2 as the building is square-shaped.

Dividing 228 by 2, we get 114.

So half of the building measures 114 square feet.

Well explained 👍

Problem 3

Calculate √228 x 5.

Okay, lets begin

75.5

Explanation

The first step is to find the square root of 228, which is approximately 15.10.

The second step is to multiply 15.10 by 5.

So 15.10 x 5 = 75.5

Well explained 👍

Problem 4

What will be the square root of (200 + 25)?

Okay, lets begin

The square root is 15

Explanation

To find the square root, we need to find the sum of (200 + 25).

200 + 25 = 225, and then 225 = 15.

Therefore, the square root of (200 + 25) is ±15.

Well explained 👍

Problem 5

Find the perimeter of the rectangle if its length ‘l’ is √200 units and the width ‘w’ is 20 units.

Okay, lets begin

We find the perimeter of the rectangle as 84 units.

Explanation

Perimeter of the rectangle = 2 × (length + width).

Perimeter = 2 × (√200 + 20)

≈ 2 × (14.14 + 20)

≈ 2 × 34.14

≈ 68.28 units.

Well explained 👍

FAQ on Square Root of 228

1.What is √228 in its simplest form?

The prime factorization of 228 is 2 x 2 x 3 x 19, so the simplest form of √228 = √(2 x 2 x 3 x 19).

2.Mention the factors of 228.

Factors of 228 are 1, 2, 3, 4, 6, 12, 19, 38, 57, 76, 114, and 228.

3.Calculate the square of 228.

We get the square of 228 by multiplying the number by itself, that is, 228 x 228 = 51984.

4.Is 228 a prime number?

5.228 is divisible by?

228 has many factors; those are 1, 2, 3, 4, 6, 12, 19, 38, 57, 76, 114, and 228.

Important Glossaries for the Square Root of 228

  • Square root: A square root is the inverse of a square. Example: 4² = 16, and the inverse of the square is the square root, that is, √16 = 4.
     
  • Irrational number: An irrational number is a number that cannot be written in the form of p/q, where q is not equal to zero and p and q are integers.
     
  • Prime factorization: Prime factorization is the process of expressing a number as the product of its prime factors.
     
  • Long division method: A technique used to find the square root of non-perfect squares by dividing the number into smaller, more manageable parts.
     
  • Approximation method: A method used to estimate the square root of a number by identifying the closest perfect squares and using them to calculate an approximate value.

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Jaskaran Singh Saluja

About the Author

Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.